Given:
- ABC is a right angle triangle (right angled at B) inscribed in the parabola y² = 4x.
- We need to find the minimum length of the intercept cut off by the tangents at A and C to the parabola from y-axis.

To find:
- The minimum length of the intercept cut off by the tangents at A and C to the parabola from y-axis.

Solution:
1. Equation of tangent at any point (a²/4, a) on the parabola y² = 4x is y = ax + a²/4.
2. For point A on the parabola, the tangent is y = ax + a²/4.
3. Slope of tangent at A is a.
4. Equation of perpendicular bisector of AB is x = a²/4.
5. Let the point of intersection of the perpendicular bisector of AB and the y-axis be P.
6. Let the coordinates of point A be (a²/4, a) and coordinates of point C be (c, 0).
7. Slope of AC is -a/c.
8. Equation of tangent at C is y = (-a/c)x + c.
9. Slope of tangent at C is -a/c.
10. Equation of perpendicular bisector of BC is y = (a/c)x - ac/2.
11. Let the point of intersection of the perpendicular bisector of BC and the y-axis be Q.
12. Now, the length of PQ is the required intercept cut off by the tangents at A and C to the parabola from y-axis.
13. Coordinates of point P are (a²/4, 0).
14. Coordinates of point Q are (0, ac/2).
15. Length of PQ is given by
PQ = √[(0 - a²/4)² + (ac/2)²]
= √[(a⁴/16) + (a²c²/4)]
= a√[(a²/16) + (c²/4)]
16. We need to minimize PQ.
17. Let f(a,c) = a√[(a²/16) + (c²/4)].
18. We need to find minimum value of f(a,c).
19. ∂f/∂a = √[(a²/16) + (c²/4)] + (a/8) * (a²/16 + c²/4)^(-1/2) * (2a/16) = 0
20. ∂f/∂c = (ac/2) * (a²/16 + c²/4)^(-1/2) * (1/2) = 0
21. From (20), we get c = 0 or a = 0.
22. If c = 0, then the triangle ABC is degenerate and PQ = AB = a²/4.
23. If a = 0, then the triangle ABC is also degenerate and PQ = BC = c/2.
24. Hence, we need to consider the case where a and c are both non-zero.
25. From (19), we get
√[(a²/16) + (c²/